If you wrote the equation without mistake: 2*x^2*(x-2)^2 = x+1 ,
then a few simple calculations will lead you to a third order equation, and there are formulas for solving analtically a third order polynimial equation.

However, as matt_grime said, there is one root that is easy to find: just try a few numbers, or make a graphic of the function to guess what is this simple root.

Once you have found this simple root, you can indeed factor it out by division and solve the second order polynomial that still remains.